Robert F. Costantino

Transitions in population dynamics: Equilibria to periodic cycles to aperiodic cycles

1. We experimentally set adult mortality rates, μ a , in laboratory cultures of the flour beetle Tribolium at values predicted by a biologically based, nonlinear mathematical model to place the cultures in regions of different asymptotic dynamics. 2. Analyses of time-series residuals indicated that the stochastic stage-structured model described the data quite well. Using the model and maximum-likelihood parameter estimates, stability boundaries and bifurcation diagrams were calculated for two genetic strains. 3. The predicted transitions in dynamics were observed in the experimental cultures. The parameter estimates placed the control and μ a = 0.04 treatments in the region of stable equilibria. As adult mortality was increased, there was a transition in the dynamics. At μ a = 0.27 and 0.50 the populations were located in the two-cycle region. With μ a = 0.73 one genetic strain was close to a two-cycle boundary while the other strain underwent another transition and was in a region of equilibrium. In the μ a = 0.96 treatment both strains were close to the boundary at which a bifurcation to aperiodicities occurs; one strain was just outside this boundary, the other just inside the boundary. 4. The rigorous statistical verification of the predicted shifts in dynamical behaviour provides convincing evidence for the relevance of nonlinear mathematics in population biology.

An interdisciplinary approach to understanding nonlinear ecological dynamics

Abstract We describe a research program which covers a spectrum of activities essential to testing nonlinear population theory: from the translation of the biology into the formal language of mathematics, to the analysis of mathematical models, to the development and application of statistical techniques for the analysis of data, to the design and implementation of biological experiments. The statistical analyses, mathematics, and biology are thoroughly integrated. We review several aspects of our current research effort that demonstrate this integration.

Power spectra reveal the influence of stochasticity on nonlinear population dynamics

Stochasticity alters the nonlinear dynamics of inherently cycling populations. The power spectrum can describe and explain the impacts of stochasticity. We fitted models to short observed time series of flour beetle populations in the frequency domain, then used a well fitting stochastic mechanistic model to generate detailed predictions of population spectra. Some predicted spectral peaks represent periodic phenomena induced or modified by stochasticity and were experimentally confirmed. For one experimental treatment, linearization theory explained that these peaks represent overcompensatory decay of deviations from deterministic oscillation. In another treatment, stochasticity caused frequent directional phase shifting around a cyclic attractor. This directional phase shifting was not explained by linearization theory and modified the periodicity of the system. If field systems exhibit directional phase shifting, then changing the intensity of demographic or environmental noise while holding constant the structure of the noise can change the main frequency of population fluctuations.

A chaotic attractor in ecology: theory and experimental data

Chaos has now been documented in a laboratory population. In controlled laboratory experiments, cultures of flour beetles (Tribolium castaneum) undergo bifurcations in their dynamics as demographic parameters are manipulated. These bifurcations, including a specific route to chaos, are predicted by a well-validated deterministic model called the “LPA model”. The LPA model is based on the nonlinear interactions among the life cycle stages of the beetle (larva, pupa and adult). A stochastic version of the model accounts for the deviations of data from the deterministic model and provides the means for parameterization and rigorous statistical validation. The chaotic attractor of the deterministic LPA model and the stationary distribution of the stochastic LPA model describe the experimental data in phase space with striking accuracy. In addition, model-predicted temporal patterns on the attractor are observed in the data. This paper gives a brief account of the interdisciplinary effort that obtained these results.

Colour of environmental noise affects the nonlinear dynamics of cycling, stage‐structured populations

Populations fluctuate because of their internal dynamics, which can be nonlinear and stochastic, and in response to environmental variation. Theory predicts how the colour of environmental stochasticity affects population means, variances and correlations with the environment over time. The theory has not been tested for cycling populations, commonly observed in field systems. We applied noise of different colours to cycling laboratory beetle populations, holding other statistical properties of the noise fixed. Theory was largely validated, but failed to predict observations in sufficient detail. The main period of population cycling was shifted up to 33% by the colour of environmental stochasticity. Noise colour affected population means, variances and dominant periodicities differently for populations that cycled in different ways without noise. Our results show that changes in the colour of climatic variability, partly caused by humans, may affect the main periodicity of cycling populations, possibly impacting industry, pest management and conservation.

Nonlinear Stochastic Population Dynamics: The Flour Beetle Tribolium as an Effective Tool of Discovery

When observation and theory collide, scientists turn to carefully designed experiments for resolution. Their motivation is especially high in the case of biological systems, which are typically far too complex to be grasped by observation and theory alone. The best procedure, as in the rest of science, is first to simplify the system, then to hold it more or less constant while varying the important parameters one or two at a time to see what happens. —Edward O. Wilson (2002)

Species competition: uncertainty on a double invariant loop

The Tribolium (flour beetle) competition experiments conducted by Park have been highly influential in ecology. We have previously shown that the dynamics of single-species Tribolium populations can be well-described by the discrete-time, 3-dimensional larva–pupa–adult (LPA) model. Motivated by Parks experiments, we explore the dynamics of a 6-dimensional “competition LPA model” consisting of two LPA models coupled through cannibalism. The model predicts a double-loop coexistence attractor, as well as an unstable exclusion equilibrium with a 5-dimensional stable manifold that plays an important role in causing one of the species to go extinct in the presence of stochastic perturbations. We also present a stochastic version of the model, using binomial and Poisson distributions to describe the aggregation of demographic events within life stages. A novel “stochastic outcome diagram,” the stochastic counterpart to a bifurcation diagram, summarizes the model-predicted dynamics of uncertainty on the double-loop. We hypothesize that the model predictions provide an explanation for Parks data. This “stochastic double-loop hypothesis” is accessible to experimental verification.

On the rate of genetic adaptation under natural selection.

Abstract For a one locus n-allele genetic model of natural selection two theoretical predictions are obtained: first, that the non-equilibrium average fitness value, W(t) is always between the equilibrium value, W∗, and W∗−H(p o , p∗) t until finally W(t) = W∗ and, secondly, that the area bounded by W∗ and the curve W(t) is equal to H(p 0 ,p∗ ), the entropy distance between initial (p0) and equilibrium ( p∗ ) gene frequency distributions. The experimental observations corresponding to these predictions are discussed.

Evolution of Corn Oil Sensitivity in the Flour Beetle

We explore the persistence of corn oil sensitivity in a population of the flour beetle Tribolium castaneum using evolutionary game methods that model population dynamics and changes in the mean strategy of a population over time. The strategy in an evolutionary game is a trait that affects the fitness of the organisms. Corn oil sensitivity represents such a strategy in the flour beetle. We adapt an existing model of the ecological dynamics of T. castaneum into an evolutionary game framework to explore the persistence of corn oil sensitivity in the population. The equilibrium allele frequencies resulting from the evolutionary game are evolutionarily stable strategies and compare favorably with those obtained from the experimental data.

An experimental check of fitness entropy vs. selective delay

Abstract A new macroparameter characterizing the movement of a non-equilibrium initial population structure toward the equilibrium structure under the natural selection process was suggested in a previous paper (Ginzburg, 1977). This parameter, named selective delay, is directly measurable from the population size growth curve without any knowledge of the frequencies of the different genotypes in the population. An equation was proposed describing the relationship between selective delay, average population fitness at the equilibrium point and the entropy distance between initial and equilibrium states of the population. The purpose of this paper is an experimental check of this equation. Experimental results with populations of flour beetle Tribolium castaneum segregating at the unsaturated fatty acid sensitive locus have shown a good correspondence to the theory.